We use the Boson-Fermion correspondence for $S$ and $Q$ functions to establish some interesting properties concerning outer products and plethysms of $S$-functions (or $Q$-functions) by power sum symmetric functions. The techniques which are developed are also applied to computing the inverse Kostka-Foulkes matrix (which is the transition matrix between Hall-Littlewood symmetric functions and $S$-functions) in some simple cases.